Outline

What is the meaning of the notion "dynamic DNAWhat is the meaning of the notion "dynamic DNA““??

Qualitative analysis of the effect of DNA dynamics on charge traQualitative analysis of the effect of DNA dynamics on charge transfer.nsfer.

TightTight--binding models and the possibility to study charge transfer beyobinding models and the possibility to study charge transfer beyond nd the Condon approximation.the Condon approximation.

Effects that exist in dynamic DNAEffects that exist in dynamic DNA

Static and Dynamic DNA StructuresStatic and Dynamic DNA Structures

RiseShift

Slide

Roll

TwistTilt

5'

3'

3'

5'

Ideal structure: Rise=3.38 Å, twist=36º,other parameters are assumed to be zero

Real structure (X-ray data for 400 base pairs): Rise=3.2÷3.6 Å, Shift=-1.0÷1.6 Å , Slide=-2.4÷2.8 Å Twist=20÷41 º, Tilt=-7.8÷6.6 º, Roll=-8.6÷25 º

Static and Dynamic DNA StructuresIs Dynamics Important indeed?

Incoherent (hopping) transport

Donor Acceptorεb εb εb εb

V12 V23 V34 VN-1, N

G+ A A A A G, GG, GGG

•••

EEDD

EEAA

εεbb

ΔεΔεWill dynamics affect the rate of elementary hopping steps?

Static and Dynamic DNA StructuresCommon Description of Elementary Hopping Step

Squared transition matrix element can be factored into an electronic part and a nuclear

part (Frank-Condon factor)

Electronic coupling, V, between D and A is independent of nuclear coordinates Q

(Condon approximation)

22π ρ=a eff FCw V 22π ρ=a FCw V22π ρ=a eff FCw V

Electronic coupling, V, between D and A is depends on nuclear coordinates Q

parametrically

Static and Dynamic DNA StructuresQualitative Analysis of non-Condon Torsional Effects

( )21 +∞

−∞

= ∫ exp ( ) ( )CT DA V FCk dt iω t C t C t

2στ

⎛ ⎞= − +⎜ ⎟

⎝ ⎠( ) exp ( )V V

rot

tC t V t

( ) ( )0 0+∞ +∞

−∞ −∞

∝ ≈ ≅∫ ∫exp ( ) ( ) ( ) exp ( ) ( ) .CT DA V FC V DA FC V FCk dt iω t C t C t C dt iω t C t C ρ

τrot << τCT

τCT>>τrot >> τFC

CV(t) is the electronic coupling correlation function and CFC(t) is the time-dependent Franck-Condon factor with the decay time τFC

with the finite time-averaged value ⟨V(t)⟩=V0 and the mean-square deviation of electronic coupling σV

22 2 200σ = − = −( ) ( )V V V t V V

θ

θ θ θ= ∫( ) ( , ) ( )CTk t P t k d

P(θ,t) is the probability density to find the two adjacent molecular subunits in the conformation with the torsion angle between θ and θ+dθ at time t

τrotτCT <

Static and Dynamic DNA StructuresImportant Limits

Fast torsional motion τCT>> τrot >> τFC and small structural fluctuation σV<<⟨V(t)⟩2= V0

20

2π ρ=CT FCk V

Fast torsional motion τCT>> τrot >> τFC and large structural fluctuation σV>>⟨V(t)⟩2= V0

22π ρ=CT FCk V

θ

θ θ θ= ∫( ) ( , ) ( )CTk t P t k d

Static nonStatic non--Condon effectCondon effect

Dynamic nonDynamic non--Condon effectCondon effect

MarcusMarcus--HushHush--Jortner equationJortner equation

Slow torsional motion τ rotτCT <

Computational Approach

The wave function of the charge,The wave function of the charge,ΨΨ, is written as a linear combination of orbitals , is written as a linear combination of orbitals localized on each sitelocalized on each site

1

( ) ( )N

i ii

t c t=

Ψ =∑ ϕ with with ci(t=0)=δ1,i

Charge motion is treated quantum mechanicallyCharge motion is treated quantum mechanically

( ) ( )ti H tt

∂Ψ= Ψ

∂

Bridge dynamics is treated classicallyBridge dynamics is treated classically

( )rot tor ii dif

B i

D U tk T

∂Δ =− Δ +Δ

∂θθ θ

θ

χχ ∈∈ [[--1/2,1/2)1/2,1/2)wherewhere andand ΔθΔθdifdif =(24=(24DDrotrotΔΔtt))1/21/2χχ ,,DDrotrot==1/(21/(2ττrotrot))

11 12

21

0 0

0

0

b

NN

VV

H

i

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟

=⎜ ⎟⎜ ⎟⎜ ⎟

−⎜ ⎟⎝ ⎠

εε

ετ

Computational Verification for Static (a and b) and Dynamic (c) Torsional Disorder

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Static disorder limit Static disorder limit ττrotrot//ττCTCT>>1>>1

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Fast fluctuation limit Fast fluctuation limit ττrotrot//ττCTCT<<1<<1

c

ττrotrot//ττCTCT =50 (1), 10 (2), 1 (3), 0.1 =50 (1), 10 (2), 1 (3), 0.1 (4), 0.01 (5)(4), 0.01 (5)

σσVV of the charge transfer integral with the average value of 0.3 eVof the charge transfer integral with the average value of 0.3 eV is 3.61is 3.61⋅⋅1010--22 eVeV22 (1), 3.28(1), 3.28⋅⋅1010--22 eVeV22 (2), 2.89(2), 2.89⋅⋅1010--22 eVeV22

(3), 2.44(3), 2.44⋅⋅1010--2 2 eVeV22 (4); (4); ΔεΔε=0.6 eV=0.6 eV

Y. A. Berlin et al. J. Phys. Chem. C Y. A. Berlin et al. J. Phys. Chem. C 112112, 10988 (2008), 10988 (2008)

Examples DNA Hairpins

Hole acceptor

Hole donor

Electronic Coupling in Dynamic DNA Hairpins

Coupling fluctuates on subCoupling fluctuates on sub--ps timescaleps timescale

AA--A coupling peaks at 0.05 eV and is larger A coupling peaks at 0.05 eV and is larger than similar coupling in static Bthan similar coupling in static B--DNA DNA structurestructure

Number of AT base pairs, Number of AT base pairs, NNATAT, does not , does not significantly influence the distribution of the significantly influence the distribution of the values of values of VVeffeff between neighboring adenines between neighboring adenines if if NNATAT≤≤66

Couplings between Sa and A and between Couplings between Sa and A and between SdSd and A are small (0.005 eV and 0.02 eV, and A are small (0.005 eV and 0.02 eV, respectively) while the meanrespectively) while the mean--square square deviation is about 0.06 eV. These findings deviation is about 0.06 eV. These findings suggest the importance of deviations from the suggest the importance of deviations from the average average VVeffeff values for charge injection values for charge injection process.process.

F. C. Grozema et al. J. Am. F. C. Grozema et al. J. Am. Chem. Soc.Chem. Soc.130130, 5157 (2008), 5157 (2008)

Site Energies in Dynamic DNA

MeanMean--square deviation in the square deviation in the fluctuating value of the adeninefluctuating value of the adenine site site energy is 0.15 eVenergy is 0.15 eVSite energies gradually increase with Site energies gradually increase with distance (number of AT base pairs)distance (number of AT base pairs)The energy barrier for hole injection The energy barrier for hole injection from Sa* into the first adenine was from Sa* into the first adenine was found to be found to be ≤≤0.4 eV0.4 eV

F. C. Grozema et al. J. Am. F. C. Grozema et al. J. Am. Chem. Soc.Chem. Soc.130130, 5157 (2008), 5157 (2008)

Charge Transfer Simulations

P(t) = cn (t)2

n

N

∑

Survival probability:

Charge spreads from first site over the bridgeCharge spreads from first site over the bridge

Charge decays irreversibly at acceptorCharge decays irreversibly at acceptor

Total population decays in timeTotal population decays in time

Hole Population as a Function of Time and ChargeDistribution in Dynamic DNA Hairpins

Population on donor,Population on donor,

Population on all bridge sites, Population on all bridge sites,

Total population = Survival probabilityTotal population = Survival probability

Hairpin 1

Hairpin 2

Hairpin 6

21 ( )c t

−

=∑

12

2

( )N

nnc t

Hairpin 1 after 1 ps Hairpin 6 after 50 ps

Kinetics of hole transfer. Effects of Electrostatic Interaction and Structural Fluctuations

F. C. Grozema et al. J. Am. F. C. Grozema et al. J. Am. Chem. Soc.Chem. Soc.130130, 5157 (2008), 5157 (2008)

Calculations:Calculations:

Experiment:Experiment: F. D. Lewis et al. J. Am. F. D. Lewis et al. J. Am. Chem. Soc.Chem. Soc.128128, 791 (2006), 791 (2006)

F. C. Grozema et al. F. C. Grozema et al. AngewAngew. . Chem., Int. Ed. Chem., Int. Ed. 4545, 7982 , 7982 (2006)(2006)

1

2

3

Curve 1:Curve 1: No fluctuations, electrostatic interaction No fluctuations, electrostatic interaction is includedis included

Curve 2:Curve 2: Both fluctuations and electrostatic Both fluctuations and electrostatic interactions are includedinteractions are included

Curve 3:Curve 3: No electrostatic interaction, fluctuations No electrostatic interaction, fluctuations are includedare included

Conclusions

The extensive study of hole transfer in dynamic DNA hairpins revThe extensive study of hole transfer in dynamic DNA hairpins reveals two important eals two important factors that affect charge motion, but so far did not receive thfactors that affect charge motion, but so far did not receive the attention they e attention they deserve in theoretical investigations of hole transport through deserve in theoretical investigations of hole transport through DNA with static DNA with static structure. One factor is structural fluctuations which change thstructure. One factor is structural fluctuations which change the values of electronic e values of electronic coupling and site energies. Another is the electrostatic interaccoupling and site energies. Another is the electrostatic interaction between a hole tion between a hole and donor anion giving rise to the barrier for charge propagatioand donor anion giving rise to the barrier for charge propagation through DNA.n through DNA.

Our analysis shows that the inclusion of these factors in the foOur analysis shows that the inclusion of these factors in the formalism of electron rmalism of electron transfer requires to consider the process of charge transition ftransfer requires to consider the process of charge transition from donor to rom donor to acceptor beyond the Condon approximation. It was demonstrated thacceptor beyond the Condon approximation. It was demonstrated that this can be at this can be done analytically in two important limits of slow and fast fluctdone analytically in two important limits of slow and fast fluctuations. uations.

Based on the computational results obtained, we propose a simpleBased on the computational results obtained, we propose a simple tighttight--binding binding model that allows the qualitative description of recent kineticsmodel that allows the qualitative description of recent kinetics data on hole transfer data on hole transfer in DNA hairpins without fitting parameters.in DNA hairpins without fitting parameters.

We also show that in short AT tracks with less than 4 base pairsWe also show that in short AT tracks with less than 4 base pairs, charge transfer , charge transfer from the hole donor to the capped stilbene acceptor occurs via tfrom the hole donor to the capped stilbene acceptor occurs via tunneling through unneling through the electrostatic barrier between donor and acceptor subunits, wthe electrostatic barrier between donor and acceptor subunits, while for longer AT hile for longer AT tracks the process can be viewed as fluctuationtracks the process can be viewed as fluctuation--assisted incoherent hoppingassisted incoherent hopping..

Acknowledgments

Prof. Mark RatnerProf. Mark Ratner

Delft University of Technology, Delft University of Technology, The Netherlands:The Netherlands:

Prof. Ferdinand C. GrozemaProf. Ferdinand C. Grozema

Prof. Laurens D.A. SiebbelesProf. Laurens D.A. Siebbeles Prof. George SchatzProf. George Schatz

Northwestern University, Northwestern University, Evanston, IL, USA:Evanston, IL, USA:

Dr. Stefano Dr. Stefano TonzaniTonzani

Acknowledgments

This work supported by the Dutch Foundation of Scientific Research (NWO) through an NWO-VENI grant to Ferdinand Grozema. The National Science Foundation (NSF) is acknowledged for financial support through the grant CHE-0628130 to George Schatz. Mark Ratner and Yuri Berlin thanks theChemistry Divisions of the NSF and ONR for financial support.